241 
determining the dispersive ratio of glass, &c. 
equally rigid with the other three, the workman is restricted 
to a very exact accordance in the measure of all his four 
tools, and it leaves him no opportunity of matching a good 
flint lens with a plate, or a plate with a flint, which is in 
many cases a desirable convenience. I have not therefore 
insisted rigidly upon a fourth condition, but have made this 
subservient to the above convenience, by only requiring that 
the contact surfaces shall be either exactly, or nearly equal, 
and the concave the deeper, when there is any difference 
between them. The optician is thus enabled to make a choice, 
within certain limits, of the radii of one of his lenses, and has 
only to match the other to it. By this means the intricate 
equation arising out of the fourth condition is avoided. I am 
quite aware, that in this way a great sacrifice is made of ana¬ 
lytical elegance; but as my object has been to bring the 
calculation fully within the reach of such practical opticians 
as have no pretensions to a knowledge of analysis, I have 
prefered a simple, although somewhat indirect method of 
computation, to one more direct and refined, but at the same 
time more intricate and laborious. The principle here pro¬ 
posed will be illustrated in the following paragraphs. 
The investigation of the aberration produced at one spheri¬ 
cal surface is found in most of our optical treatises, and 
need not therefore be repeated in this place; of these ex¬ 
pressions, that which is given in Wood's Optics (art. 397) is 
perhaps one of the most simple. I shall therefore adopt this, 
and refer the reader to the work itself for the investigation. 
Let d be the distance of a radiant point from a spherical 
convex surface of a denser medium whose radius is r, and 
semidiameter y ; let also the sine of incidence to the sine of 
MDCCCXXVII. I i 
